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Chapter 10: Problem 62

If \(G\) is a simple graph with 15 edges and \(\overline{G}\) has 13 edges, how many vertices does \(G\) have?

Short Answer

Expert verified
Graph G has 8 vertices.

Step by step solution

01

Understand the relationship between a graph and its complement

For a simple graph and its complement, the sum of the number of edges in both graphs is equal to the number of edges in a complete graph with the same number of vertices.
02

Express the relationship using an equation

Let the number of vertices in graph G be denoted as n. A complete graph with n vertices has \(\binom{n}{2}\) edges. Thus, the equation \(\binom{n}{2} = 15 + 13 \) represents the total number of edges.
03

Solve for the number of vertices

We have the equation \(\binom{n}{2} = 28\). This simplifies to \(\frac{n(n-1)}{2} = 28\). Solving this equation, we multiply both sides by 2 to get \(n(n - 1) = 56\). Finding the integer solution to this quadratic equation, we get \(8 \) vertices, because \(8 \times 7 = 56\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Graph
In graph theory, a **simple graph** refers to an undirected graph without loops or multiple edges between the same pair of vertices. This means each edge connects a unique pair of vertices without repetition. Simple graphs are easy to understand and form the basis for more complex graph structures.
To understand them better:
  • Imagine a set of points (called vertices) connected by lines (called edges).
  • There are no extra lines connecting the same points more than once.
  • No vertex is connected to itself (no loops).
These properties help keep the graph simple and straightforward, ensuring clarity in relationships between vertices.
Complement of a Graph
The **complement of a graph** \(\backslash overline{G}\) involves a transformation of the original graph G. In the complement:
  • Every pair of vertices that are connected by an edge in G are NOT connected in \(\backslash overline{G}\).
  • Every pair of vertices that are NOT connected by an edge in G are connected in \(\backslash overline{G}\).
Essentially, the edges in \(\backslash overline{G}\) represent all the edges missing in G to make it a complete graph. This understanding is crucial for solving problems involving the relationship between a graph and its complement.
Complete Graph
A **complete graph** is a simple graph in which every pair of distinct vertices is connected by a unique edge. Imagine a network where everyone knows everyone else. This type of graph is critical to understand because:
  • It contains the maximum number of edges possible for a given number of vertices.
  • If the graph has n vertices, the number of edges is \(\backslash binom{n}{2}\), which equals \(\frac{n(n-1)}{2}\).
This formula helps in deriving the relationships and counts of edges when considering complements and other graph properties.
Vertices
In graph theory, **vertices** (or nodes) are the fundamental units used to build graphs. They represent points or locations that can be connected by edges. Understanding vertices involves noting:
  • In a graph G, the number of vertices is denoted as **n**.
  • Vertices can be connected by edges to form various structures and shapes.
  • The degree of a vertex is the number of edges incident to it.
Each vertex in a simple graph represents a unique entity that can be connected to other vertices without any repetitions or loops.
Edges
An **edge** in graph theory is a line connecting two vertices. These connections can define many characteristics of graphs. To understand edges, keep these points in mind:
  • In a simple graph, an edge connects two distinct vertices without repetitions.
  • An edge between vertex A and vertex B is unique in a simple graph.
  • In a complete graph, the total number of edges is given by \(\backslash binom{n}{2}\), where n is the number of vertices.
Edges play a crucial role in defining the structure and properties of graphs, making them essential for graph analysis and understanding relationships within data.

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Most popular questions from this chapter

Fleury's algorithm, published in \(1883,\) constructs Euler circuits by first choosing an arbitrary vertex of a connected multigraph, and then forming a circuit by choosing edges are successively. Once an edge is chosen, it is removed. Edges are chosen successively so that each edge begins where the last edge ends, and so that this edge is not a cut edge unless there is no alternative. Prove that Fleury's algorithm always produces an Euler circuit.

Suppose there is an integer k such that every man on a desert island is willing to marry exactly k of the women on the island and every woman on the island is willing to marry exactly k of the men. Also, suppose that a man is willing to marry a woman if and only if she is willing to marry him. Show that it is possible to match the men and women on the island so that everyone is matched with someone that they are willing to marry.

Show that if \(G\) is a connected graph, then it is possible to remove vertices to disconnect \(G\) if and only if \(G\) is not a complete graph.

Suppose that a connected planar graph has eight vertices, each of degree three. Into how many regions is the plane divided by a planar representation of this graph?

Schedule the final exams for Math 115, Math 116, Math 185, Math 195, CS 101, CS 102, CS 273, and CS 473, using the fewest number of different time slots, if there are no students taking both Math 115 and CS 473, both Math 116 and CS 473, both Math 195 and CS 101, both Math 195 and CS 102, both Math 115 and Math 116, both Math 115 and Math 185, and both Math 185 and Math 195, but there are students in every other pair of courses.
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